Constructing digitized representations of objects, with the emphasis on precision and efficiency (either by means of synthesis, see, for example, Bresenham's line algorithm or digital disks, or by means of digitization and subsequent processing of digital images).

Transforming digitized representations of objects, for example (A) into simplified shapes such as (i) skeletons, by repeated removal of simple points such that the digital topology of an image does not change, or (ii) medial axis, by calculating local maxima in a distance transform of the given digitized object representation, or (B) into modified shapes using mathematical morphology.

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A 2D digital space usually means a 2D grid space that only contains integer points in 2D Euclidean space. A 2D image is a function on a 2D digital space (See image processing).

In Rosenfeld and Kak's book, digital connectivity are defined as the relationship among elements in digital space. For example, 4-connectivity and 8-connectivity in 2D. Also see pixel connectivity. A digital space and its (digital-)connectivity determine a digital topology.

A digitally continuous function means a function in which the value (an integer) at a digital point is the same or off by at most 1 from its neighbors. In other words, if x and y are two adjacent points in a digital space, |f(x) − f(y)| ≤ 1.

A gradually varied function is a function from a digital space Σ{\displaystyle \Sigma } to {A1,…,Am}{\displaystyle \{A_{1},\dots ,A_{m}\}} where A1<⋯<Am{\displaystyle A_{1}<\cdots <A_{m}} and Ai{\displaystyle A_{i}} are real numbers. This function possesses the following property: If x and y are two adjacent points in Σ{\displaystyle \Sigma }, assume f(x)=Ai{\displaystyle f(x)=A_{i}}, then f(y)=Ai{\displaystyle f(y)=A_{i}}, f(x)=Ai+1{\displaystyle f(x)=A_{i+1}}, or Ai−1{\displaystyle A_{i-1}}. So we can see that the gradually varied function is defined to be more general than the digitally continuous function.

An extension theorem related to above functions was mentioned by A. Rosenfeld (1986) and completed by L. Chen (1989). This theorem states: Let D⊂Σ{\displaystyle D\subset \Sigma } and f:D→{A1,…,Am}{\displaystyle f:D\rightarrow \{A_{1},\dots ,A_{m}\}}. The necessary and sufficient condition for the existence of the gradually varied extension F{\displaystyle F} of f{\displaystyle f} is : for each pair of points x{\displaystyle x} and y{\displaystyle y} in D{\displaystyle D}, assume f(x)=Ai{\displaystyle f(x)=A_{i}} and f(y)=Aj{\displaystyle f(y)=A_{j}}, we have |i−j|≤d(x,y){\displaystyle |i-j|\leq d(x,y)}, where d(x,y){\displaystyle d(x,y)} is the (digital) distance between x{\displaystyle x} and y{\displaystyle y}.